Using the first principle find the derivatives of Cos x
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Step-by-step solution
6 stepsLet's find the derivative of cos(x) using the first principle.
Write down the first principle formula for the derivative.
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
Substitute f(x) = cos(x) into the formula.
f'(x) = lim(h→0) [cos(x + h) - cos(x)] / h
Use the trigonometric identity cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
Here, A = x and B = h.
f'(x) = lim(h→0) [cos(x)cos(h) - sin(x)sin(h) - cos(x)] / hRearrange the terms to group cos(x) and sin(x).
f'(x) = lim(h→0) [cos(x)(cos(h) - 1) - sin(x)sin(h)] / h
f'(x) = lim(h→0) [cos(x)(cos(h) - 1)/h - sin(x)sin(h)/h]Apply the limit.
We use the standard limits:
lim(h→0) (cos(h) - 1) / h = 0
lim(h→0) sin(h) / h = 1
Since cos(x) and sin(x) do not depend on h, they are treated as constants in the limit.
f'(x) = cos(x) * lim(h→0) [(cos(h) - 1)/h] - sin(x) * lim(h→0) [sin(h)/h]
f'(x) = cos(x) * 0 - sin(x) * 1
f'(x) = 0 - sin(x)Simplify to get the final derivative.
f'(x) = -sin(x)
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